An online probabilistic prediction method for bridge bearing displacement
Through flow sparse Gaussian process model and variational inference, the problem of time-varying model parameters in bridge bearing displacement prediction is solved, and efficient and accurate real-time prediction and uncertainty interpretation are achieved, which is suitable for bridge health monitoring.
Patent Information
- Application Number
- CN202411750263.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-02
- Publication Date
- 2025-08-22
- Estimated Expiration
- 2044-12-02
AI Technical Summary
The prior art is difficult to consider the time-change of model parameters caused by material aging and service environment changes in the bridge support displacement prediction, resulting in a decrease in prediction accuracy and unable to achieve accurate long-term prediction.
The flow sparse Gaussian process model is adopted, and the approximate prior distribution is updated by introducing pseudo-points, combining variational inference and Bayesian theorem, an online learning framework is built, and the model parameters are gradually updated to realize real-time probability prediction of support displacement.
It improves the computing efficiency of model training and prediction, reduces storage requirements, ensures prediction accuracy under the aging of the bearing and changes in the service environment, can update model parameters in real time, and provide predicted mean and uncertainty explanations.
Smart Images
Figure CN119578247B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of bridge service status assessment, and in particular to an online probability prediction method for bridge support displacement. Background Art
[0002] The bearing is an important component that connects the upper and lower structures of the bridge and transfers loads. Its health status affects the operational safety of the bridge. Under the influence of the environment and vehicle loads, the daily cumulative displacement of the bearing continues to increase, resulting in deterioration of the bearing performance and defects such as excessive wear. Deterioration of bearing performance can easily lead to restricted relative movement between the main beam and the pier, and the bridge structure is subjected to large additional internal forces, which can cause defects such as cracking of the main beam. Bearing displacement is one of the health monitoring indicators of long-span bridges. The operating status of the bearing can be evaluated by comparing the difference between the measured bearing displacement under the action of ambient temperature and the theoretically predicted displacement. Therefore, establishing an accurate ambient temperature-bearing displacement model and evaluating its service performance are crucial to ensuring the safe operation of bridge structures and extending their service life.
[0003] Existing studies often use deterministic models to predict the displacement response of bearings under temperature, which makes it difficult to consider the uncertainty of monitoring data. Probabilistic prediction models estimate target values in the form of probability distributions and provide more information in the form of probability density, quantiles, and prediction intervals. Currently, most methods for predicting the probability of bearing displacement are based on offline learning algorithms that assume that the model coefficients are constant, that is, the optimal values of the model parameters are estimated once on the training data set. However, the aging of bearing materials and changes in the service environment can easily cause the parameters of the bearing displacement prediction model to vary over time. Offline learning algorithms can only guarantee prediction accuracy within a certain time range and cannot achieve accurate prediction of bridge bearing displacement during service. The Gaussian process model is a non-parametric supervised learning model with a specific probabilistic interpretation under the Bayesian framework. It can be extended to an online learning framework by combining Bayesian inference. Therefore, how to combine the online Gaussian process model and bridge health monitoring stream data to gradually update the model parameters and propose a real-time probabilistic prediction method for bridge bearing displacement is a difficult problem that needs to be solved urgently. Summary of the Invention
[0004] In order to overcome the deficiencies in the prior art, the present invention provides an online probability prediction method for bridge bearing displacement to solve the problem that the displacement changes of bridge bearings during long-term operation are difficult to accurately predict due to material aging and changes in the service environment.
[0005] To achieve the above object, the present invention adopts the following technical solutions:
[0006] A bridge support displacement online probability prediction method includes the following steps:
[0007] Step 1: Use the standard Gaussian process model as the baseline probability prediction model: y n=f(x n )+ε n , where ε n is the noise term, is the variance; f is the training function, and the training function f obeys the Gaussian process prior distribution with mean 0 and covariance k: Where θ is the hyperparameter on which the covariance k depends and The set of hyperparameters composed of
[0008] Definition: Given a training function f and variance After that, when the main beam temperature input vector is X, the support displacement output vector of the probability prediction model is y;
[0009] Step 2: Obtain historical girder temperature monitoring data through the bridge health monitoring system and historical monitoring data of bearing displacement according to Introducing a set of pseudo points in u i =f(z i ), M<<N, M is the number of pseudo points, N is the number of samples; Definition: the set of training function values of pseudo point P The training function value set of historical monitoring data The remaining training function value set f ≠u =fu;
[0010] According to u and f, f ≠u The relationship between , the product rule is used to obtain the true prior distribution of the training function f: About the expansion of u, and according to Get the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u, and get the expansion of the approximate joint distribution p(f,y|θ) of the training function f and the output vector y with respect to u based on the original likelihood function of p(f|θ) and the output vector y;
[0011] Define the true prior distribution of u under the output vector y as p(u|y,θ). According to p(u|y,θ) and the relationship between u and f, f ≠u The product rule is used to obtain the expansion of the true posterior distribution p(f|y,θ) of the training function f under the output vector y with respect to u, and the expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained based on p(f|y,θ);
[0012] It should be noted that in step 2, u is f={f ≠u ,u}, satisfying u=ff ≠u , according to u and f, f ≠uThe relationship between the true prior distribution of the training function f The expansion of u is obtained by using the product rule according to formula (2.1):
[0013]
[0014] Where: p(u|θ) is the prior distribution of u; p(f|u,θ) is the prior distribution of f under u; p(f ≠u |f,u,θ) is f ≠u Prior distribution under the conditions of f and u;
[0015] Among them, p(u|θ) is determined according to formula (2.2):
[0016]
[0017] Where: K uu is the covariance matrix between u and u;
[0018] Among them, p(f|u,θ) is determined according to formula (2.3):
[0019]
[0020] Where: K fu is the covariance matrix between f and u; K ff is the covariance matrix between f and f; K uf is the covariance matrix between u and f; K uu The inverse matrix of .
[0021] It should be noted that in step 2, in order to reduce K fu and K ff To reduce the complexity of the derivation process, we define the approximate posterior distribution q(f|u,θ) of the training function value set f under the training function value set u of the pseudo point P as the approximate p(f|u,θ), that is, q(f|u,θ)≈p(f|u,θ); since u and f ≠u are independent of each other, and u is independent of f and f ≠u u=ff ≠u Relationship, that is, p(f ≠u |u,θ)≈p(f ≠u |f,u,θ); q(f|u,θ), p(f ≠u |u,θ) replace p(f|u,θ), p(f ≠u |f,u,θ), the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u is obtained according to formula (2.4):
[0022] p(f|θ)=p(u|θ)q(f|u,θ)p(f ≠u |u,θ) (2.4)
[0023] Where: p(f|θ) is the approximate prior distribution of the training function f; q(f|u,θ) is the approximate posterior distribution of the training function value f under the condition u, which is an approximation of p(f|u,θ); p(f ≠u |u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.
[0024] It should be noted that in step 2, the approximate joint distribution p(y,f|θ) of the training function f and the output vector y is obtained according to formula (2.5):
[0025]
[0026] Where: p(y,f|θ) is the approximate joint distribution of the training function f and the output vector y; is the original likelihood function.
[0027] Substituting equation (2.4) into equation (2.5), the expansion of the approximate joint distribution p(y,f|θ) with respect to u is obtained according to equation (2.6):
[0028] p(y,f|θ)=p(y|f,θ)p(f ≠u |u,θ)p(u|θ) (2.6)
[0029] Where p(y|f,θ) is the prior distribution of the output vector y under the training function f.
[0030] It should be noted that in step 2, the true posterior distribution p(f|y,θ) of the training function f under the output vector y is expanded with respect to u according to formula (2.7):
[0031] p(f|y,θ)=p(u|y,θ)p(f ≠u |y,u,θ) (2.7)
[0032] Where p(f|y,θ) is the true posterior distribution of f under the condition of y; p(u|y,θ) is the prior distribution of u under the condition of y; p(f ≠u |y,u,θ) is f ≠u Prior distribution under u and y conditions;
[0033] According to p(u|y,θ) and u and f, f ≠u The relationship between the training function f and the approximate posterior distribution q(f|θ) with respect to u is obtained by formula (2.8):
[0034] q(f|θ)=p(f≠u |u,θ)q(u|θ) (2.8)
[0035] Where: q(u|θ) is the posterior distribution of u; p(f ≠u |u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.
[0036] Step 3: Define the marginal likelihood function of the output vector y as p(y|θ), and construct the lower bound of the logarithmic marginal likelihood function logp(y|θ) based on the approximate posterior distribution q(f|θ) and the approximate joint distribution p(y,f|θ), combined with the principle that p(y|θ) has an analytical solution; define the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ), and construct the variational free energy bound based on the difference between the logarithmic marginal likelihood function logp(y|θ) and the KL divergence according to the variational inference principle. As a lower bound for logp(y|θ), we can derive Expression about u; according to Select different pseudo points P and use numerical iteration method to iterate u, so that the approximate posterior distribution q(f|θ) approaches the true posterior distribution p(f|y,θ) so that Maximize and obtain the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ);
[0037] It should be noted that in step 3, the lower bound of the logarithmic marginal likelihood function logp(y|θ) is constructed according to formula (3.1):
[0038]
[0039] Where: logp(y|θ) is the logarithmic marginal likelihood function of the output vector y; q(f|θ) is the approximate posterior distribution of the training function f; is the variational free energy bound that depends on the approximate posterior distribution q(f|θ); p(y,f|θ) is the approximate joint distribution of the training function f and the output vector y;
[0040] Among them, the variational free energy bound Determine according to formula (3.2):
[0041]
[0042] Where: KL[q(f|θ)||p(f|y,θ)] is the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ); p(f|y,θ) is the true posterior distribution of the training function f under the condition of the output vector y;
[0043] Substituting equations (2.6) and (2.8) into equation (3.1), The expression of u is obtained according to formula (3.3):
[0044]
[0045] Where: f n =f(x n ) is x n The implicit function value at x n That is the main beam temperature historical monitoring data The nth sample data in ; To eliminate items.
[0046] It should be noted that in step 3, the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ) is determined according to formula (3.4) and (3.5) respectively:
[0047]
[0048] Where I is the identity matrix; trace(·) is the regularization tracking term to prevent the model from overfitting.
[0049] Step 4: According to the optimal variational free energy bound Get the value of the hyperparameter set θ, according to the optimal approximate posterior distribution q vfe (f|θ) and the hyperparameter set θ value to obtain the specific expression and variance of the training function f Substitute the specific values of into step 1 to obtain the specific expression of the benchmark probability prediction model;
[0050] Step 5: Obtain new monitoring data of main beam temperature and support displacement through the bridge health monitoring system, and compare the new monitoring data with the old historical monitoring data in step 2. Composing new historical monitoring data according to Introduce a new pseudo point P according to the method of step 2 new and P new The training function value set u new ; Based on the old true posterior distribution p(f|y,θ), use Bayes’ theorem to derive the new true posterior distribution p(f|y,y new ,θ new ); define the new approximate posterior distribution of the training function f as q new (f|θ new ), define the new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θnew ) and construct a new variational free energy bound based on the new KL divergence and the variational inference principle. According to u and u new The relationship between the two is used to derive the new variational free energy bound. About u, u new The expression of Select multiple different pseudo points P new , using numerical iteration method to calculate u new Perform iterative calculations to make the new approximate posterior distribution q new (f|θ new ) approximates the new true posterior distribution p(f|y,y new ,θ new )make Maximize and obtain the new optimal variational free energy bound and the new optimal approximate posterior distribution q new,vfe (u new |θ new );
[0051] It should be noted that in step 5, based on the old true posterior distribution p(f|y,θ) and the old optimal approximate posterior distribution q(f|θ), using Bayes’ theorem, the new true posterior distribution p(f|y,y new ,θ new ) According to formula (5.1), we can get:
[0052]
[0053] Where: q new (f|θ new ) is the new approximate posterior distribution after obtaining new monitoring data; p(f|y,y new ,θ new ) is the new true posterior distribution of the training function f after obtaining new monitoring data; is dependent on the hyperparameter θ new The normalization constant of p(f|θ new ) is the new true prior distribution of the training function f after obtaining the new monitoring data; p(y|f) is the conditional distribution of the old output vector y in the new data under the given training function f after obtaining the new monitoring data; p(y new |f) is the new output vector y in the new data given the training function f after obtaining the new monitoring data new The conditional distribution of
[0054] Among them, p(y|f) is obtained according to formula (5.2):
[0055]
[0056] Where: is a normalization constant that depends on the hyperparameter θ; q vfe (f|θ) is the optimal approximate posterior distribution before acquiring new monitoring data; To obtain the true prior distribution of the training function f before new monitoring data.
[0057] It should be noted that in step 5, the new variational free energy bound According to formula (5.3), we can get:
[0058]
[0059] Where: logp(y new |y,θ new ) is the new output vector y new The log-marginal likelihood function under the condition of the old output vector y; KL[q new (f|θ new )||p(f|y,y new ,θ new )] is the new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θ new ) new KL divergence between ;
[0060] Among them, KL[q new (f|θ new )||p(f|y,y new ,θ new )] According to formula (5.4), we can get:
[0061]
[0062] Where: p(f ≠u,new |u new ,θ new ) is the training function value set u except the new pseudo point new The remaining potential function value f ≠u,new Prior distribution of q new (u new |θ new ) is the new pseudo-point training function value set u new The posterior distribution of p(u new |θ new ) New pseudo-point training function value set u new Prior distribution of f ≠u,new =f new -u new ;f new is the set of training function values for the new historical monitoring data;
[0063] Assuming that q(f|θ) obeys the normal distribution, q(f|θ) is determined according to formula (5.5):
[0064]
[0065] Where: and are the mean and variance of the normal distribution respectively;
[0066] Among them, the new approximate posterior distribution q new (f|θ new ) The expansion of u is obtained according to formula (5.6):
[0067] q new (f|θ new )=q new (u new |θ new )p(f ≠u,new |u new ,θ new ) (5.6)
[0068] Where: p(f ≠u,new |u new ,θ new ) is the training function value set u except the new pseudo point new The remaining potential function value f ≠u,new Prior distribution of q new (u new |θ new ) is the new pseudo point u new The training function value set u new The posterior distribution of .
[0069] It should be noted that in step 5, the new optimal approximate posterior distribution q new,vfe (f|θ new ) and the new optimal variational free energy bound According to formula (5.7) and (5.8) respectively, we can obtain:
[0070]
[0071] Where: p new (u new |θ new ) is the new pseudo point P new The training function value set u new The prior distribution of K fu,new f and u new The covariance matrix between uu,new for u and u new The covariance matrix between ; K uu,new The inverse matrix of
[0072] Step 6: According to Get θ new Value, according to q new,vfe (u new |θ new ) and θ new It is worth getting the new specific expression and variance of the training function f Substitute the new specific values into step 1 to obtain a new specific expression of the baseline probability prediction model.
[0073] Compared with the prior art, the present invention has the following beneficial effects:
[0074] 1. The flow sparse Gaussian process model described in the present invention is a standard Gaussian process sparse approximate inference framework based on variational free energy. It uses pseudo-point updates to approximate the prior distribution to obtain an approximation of the true posterior distribution. It is consistent with the standard Gaussian process model, can provide a predicted mean, and explain the uncertainty caused by measurement and model errors.
[0075] 2. The time complexity of training and prediction of the flow sparse Gaussian process model described in the present invention is lower than that of the standard Gaussian process model, and avoids the low efficiency or unsolvable situation when the standard Gaussian process model processes massive bridge health monitoring data, significantly improving the computational efficiency and accuracy of model training and prediction.
[0076] 3. The present invention establishes an online learning framework based on variational inference and stream sparse Gaussian process approximation, which can perform online posterior update and hyperparameter learning in a data stream environment where monitoring data arrives sequentially, ensuring the model prediction accuracy under bearing aging and service environment changes.
[0077] 4. The present invention only needs to store a small amount of model posterior parameters and new monitoring data, which significantly reduces the model's storage requirements and dependence on data. At the same time, the present invention summarizes historical old data through a small number of pseudo points, which can avoid the computational complexity explosion problem caused by processing massive accumulated historical old data. BRIEF DESCRIPTION OF THE DRAWINGS
[0078] Figure 1 This is a flow chart of the online probability prediction method for bridge support displacement of the present invention. DETAILED DESCRIPTION
[0079] The technical solutions of the present invention will be described clearly and completely below with reference to the accompanying drawings.
[0080] like Figure 1 , Figure 1 This is a flow chart of the online probability prediction method for bridge support displacement of the present invention.
[0081] The present invention provides an online probability prediction method for bridge support displacement, comprising the following steps:
[0082] Step 1: Use the standard Gaussian process model as the baseline probability prediction model: y n =f(x n )+ε n , where f is the training function, ε n is the noise term, is the variance, assuming that the training function f obeys a Gaussian process prior distribution with mean 0 and covariance k: Where θ is the hyperparameter on which the covariance k depends and The set of hyperparameters composed of
[0083] Definition: Given a training function f and variance After that, when the main beam temperature input vector is X, the support displacement output vector of the probability prediction model is y.
[0084] The standard Gaussian process model is a conventional technology. In order to introduce symbols to facilitate the derivation of the flow sparse Gaussian process model, this patent only simply deduces the standard Gaussian process model:
[0085] Given N input and output pairs hour( Corresponding to the historical monitoring data of main beam temperature and support displacement respectively), the standard Gaussian process regression model is:
[0086] y n =f(x n )+ε n
[0087] Where f is the unknown training function; is the noise term, is the variance.
[0088] It is usually assumed that the training function f follows a Gaussian process prior distribution with mean 0 and covariance k(·,·):
[0089]
[0090] Where k(·,·) is the covariance function, k(·,·) is an unknown function containing unknown hyperparameters; θ is the hyperparameter and variance on which the covariance function k(·,·) depends. The set of hyperparameters.
[0091] Therefore, the joint distribution of the training function f and the output vector y is:
[0092]
[0093] in, is the true joint distribution of the training function f and the output vector y; is the true prior distribution of the training function f is the original likelihood function of the output vector y.
[0094] Step 2: Obtain historical girder temperature monitoring data through the bridge health monitoring system And the corresponding support displacement historical monitoring data according to Introducing a set of pseudo points in u i =f(z i ), M<<N, M is the number of pseudo points, N is the number of samples; Definition: the set of training function values of pseudo point P The training function value set of historical monitoring data The remaining training function value set f ≠u =fu;
[0095] According to u and f, f ≠u The relationship between the product rule is used to obtain the true prior distribution of the training function f About the expansion of u, and according to Get the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u, and get the expansion of the approximate joint distribution p(f,y|θ) of the training function f and the output vector y with respect to u based on the original likelihood function of p(f|θ) and the output vector y;
[0096] Define the true prior distribution of u under the output vector y as p(u|y,θ). According to p(u|y,θ) and the relationship between u and f, f ≠u The product rule is used to obtain the expansion of the true posterior distribution p(f|y,θ) of the training function f under the output vector y with respect to u, and the expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained based on p(f|y,θ);
[0097] It should be noted that in step 2, u is f={f ≠u ,u}, satisfying u=ff ≠u , according to u and f, f ≠u The relationship between the true prior distribution of the training function f The expansion of u is obtained by using the product rule according to formula (2.1):
[0098]
[0099] Where: p(u|θ) is the prior distribution of u; p(f|u,θ) is the prior distribution of f under u; p(f ≠u |f,u,θ) is f ≠u Prior distribution under the conditions of f and u;
[0100] Among them, p(u|θ) is determined according to formula (2.2):
[0101]
[0102] Where: K uu is the covariance matrix between u and u;
[0103] Among them, p(f|u,θ) is determined according to formula (2.3):
[0104]
[0105] Where: K fu is the covariance matrix between f and u; K ff is the covariance matrix between f and f; K uf is the covariance matrix between u and f; K uu The inverse matrix of
[0106] To reduce K fu and K ff To reduce the complexity of the derivation process, we define the approximate posterior distribution q(f|u,θ) of the training function value set f under the condition of the training function value set u as the approximate p(f|u,θ), that is, q(f|u,θ)≈p(f|u,θ); since u and f ≠u are independent of each other, and u is independent of f and f ≠u u=ff ≠u Relationship, that is, p(f ≠u |u,θ)≈p(f ≠u |f,u,θ); q(f|u,θ), p(f ≠u |u,θ) replace p(f|u,θ), p(f ≠u |f,u,θ), the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u is as follows (2.4):
[0107] p(f|θ)=p(u|θ)q(f|u,θ)p(f ≠u |u,θ)(2.4)
[0108] Where: p(f|θ) is the approximate prior distribution of the training function f; q(f|u,θ) is the approximate posterior distribution of the training function value f under the condition u; p(f ≠u|u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.
[0109] It should be noted that in step 2, based on the approximate prior distribution p(f|θ) of the training function f and the original likelihood function of the output vector y, the approximate joint distribution p(y,f|θ) of the training function f and the output vector y is obtained according to formula (2.5):
[0110]
[0111] Where: p(y,f|θ) is the approximate joint distribution of the training function f and the output vector y; is the original likelihood function.
[0112] Among them, combining equations (2.4) and (2.5) can rewrite the approximate joint distribution p(y,f|θ):
[0113] p(y,f|θ)=p(y|f,θ)p(f ≠u |u,θ)p(u|θ)(2.6)
[0114] Where p(y|f,θ) is the prior distribution (likelihood function) of the output vector y under the training function f.
[0115] It should be noted that in step 2, according to p(u|y,θ) and u and f, f ≠u The relationship between the true posterior distribution p(f|y,θ) and u is obtained by formula (2.7):
[0116] p(f|y,θ)=p(u|y,θ)p(f ≠u |y,u,θ)(2.7)
[0117] Where p(f|y,θ) is the true posterior distribution of f under the condition of y; p(u|y,θ) is the prior distribution of u under the condition of output vector y; p(f ≠u |y,u,θ) is the set of remaining training function values f ≠u The prior distribution conditional on u and the output vector y;
[0118] Since p(f ≠u |u,θ)≈p(f ≠u |f,u,θ), p(u|y,θ)≈q(u|θ), so the expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained according to formula (2.8):
[0119] q(f|θ)=p(f ≠u |u,θ)q(u|θ)(2.8)
[0120] Where: q(u|θ) is the posterior distribution of u; p(f ≠u |u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.
[0121] Due to historical monitoring data The number of samples is large, and the training time complexity of the standard Gaussian process model is The prediction time complexity of the test set is in: is a time complexity function, N is the number of samples, and the model training time is long. To characterize the entire historical monitoring data Construct an approximate prior distribution p(f|θ) of the training function f to replace its true prior distribution. The probability prediction model based on the approximate prior distribution p(f|θ) is the benchmark model of the stream sparse Gaussian process model established by the present invention. Its time complexity is The prediction time complexity of the test set is M is the number of pseudo points; usually M<<N, so the use of the sparse Gaussian process regression model described in this application can significantly improve the computational efficiency.
[0122] It should be noted that in step 1, a probability prediction model based on a standard Gaussian process is constructed. After approximation, the approximate joint distribution p(f,y|θ) of the training function f and the output vector y is obtained. p(f,y|θ) contains the approximate process. The probability prediction model obtained based on f in p(f,y|θ) is the approximate probability prediction model based on the flow sparse Gaussian process.
[0123] Step 3: Define the marginal likelihood function p(y|θ) of the output vector y, and construct the lower bound of the logarithmic marginal likelihood function logp(y|θ) based on the approximate posterior distribution q(f|θ) and the approximate joint distribution p(y,f|θ), combined with the principle that p(y|θ) has an analytical solution; define the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ), and construct the variational free energy bound based on the difference between the logarithmic marginal likelihood function logp(y|θ) and the KL divergence according to the variational inference principle. As a lower bound for logp(y|θ), we can derive Expression about u; according to Select different pseudo points P and use numerical iteration method to iteratively calculate u, so that the approximate posterior distribution q(f|θ) approaches the true posterior distribution p(f|y,θ) so that Maximize and obtain the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ);
[0124] It should be noted that in step 3, based on the principle that the marginal likelihood function p(y|θ) of the output vector y has an analytical solution, the hyperparameter set θ can be trained by optimizing the marginal likelihood function. According to the approximate posterior distribution q(f|θ) and the approximate joint distribution p(y,f|θ), the lower bound of the logarithmic marginal likelihood function logp(y|θ) is constructed according to formula (3.1):
[0125]
[0126] Where: logp(y|θ) is the logarithmic marginal likelihood function of the output vector y; q(f|θ) is the approximate posterior distribution of the training function f; is the variational free energy bound that depends on the approximate posterior distribution q(f|θ); p(y,f|θ) is the approximate joint distribution of the training function f and the output vector y;
[0127] Among them, the variational free energy bound It can be written as the difference between the logarithmic marginal likelihood function logp(y|θ) and the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ) of the training function f under the output vector y, that is, Determine according to formula (3.2):
[0128]
[0129] where KL[q(f|θ)||p(f|y,θ)] is the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ); and p(f|y,θ) is the true posterior distribution of the training function f under the condition of the output vector y.
[0130] It should be noted that in step 3, by substituting equations (2.6) and (2.8) into equation (3.1), a key term p(f ≠u |u,θ), a computationally tractable variational free energy bound can be derived Right now The expression of u is obtained according to formula (3.3):
[0131]
[0132] Where: f n =f(x n ) is x n The implicit function value at x n That is the main beam temperature historical monitoring data The nth sample data in ; To eliminate items.
[0133] It should be noted that in step 3, the main beam temperature z in the pseudo point P isi Define historical monitoring data of beam temperature In the main range, that is However, when selecting the pseudo point P, it is impossible to minimize the KL divergence or the variational free energy bound at one time. The maximum value is obtained by numerical iteration to determine whether the KL divergence is the minimum after a pseudo point P is selected. If the KL divergence is not the minimum, the main beam temperature z in the pseudo point P needs to be i Update and iterate u to make the approximate posterior distribution q(f|θ) approach the true prior distribution p(f|y,θ) to minimize the KL divergence. Minimizing the KL divergence is equivalent to maximizing the variational free energy bound, that is: the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ) is determined according to formula (3.4) and (3.5) respectively:
[0134]
[0135] Where I is the identity matrix; trace(·) is the regularization tracking term to prevent the model from overfitting.
[0136] Conventional Bayesian theorem normalizes the likelihood function and prior distribution to derive the posterior distribution. This method has high computational complexity, relies on the prior distribution, and can lead to unstable results due to small data sets. This patent obtains the optimal approximate posterior distribution by minimizing the KL divergence between the approximate posterior distribution and the true posterior distribution, and rigorously approximates the exact GP model by minimizing the distance between the sparse model and the exact model.
[0137] The standard Gaussian process is a non-parametric supervised learning model with a specific probabilistic interpretation under the Bayesian framework, and provides model interpretation capabilities with statistical knowledge. The stream sparse Gaussian process model described in the present invention is a standard Gaussian process approximate inference framework based on variational free energy. It uses pseudo-point updates to approximate the prior distribution to obtain an approximation of the true posterior distribution. It can not only provide a predicted mean, but also explain the uncertainty caused by measurement and model errors.
[0138] Step 4: According to the optimal variational free energy bound Get the value of the hyperparameter set θ, according to the optimal approximate posterior distribution q vfe (f|θ) and the hyperparameter set θ value to obtain the specific expression and variance of the training function f Specific values of , substitute the two into step 1 to obtain the specific expression of the probability prediction model;
[0139] Step 5: Obtain new monitoring data of main beam temperature and support displacement through the bridge health monitoring system, and compare the new monitoring data with the old historical monitoring data in step 1. Composing new historical monitoring data according to Introduce a new pseudo point P according to the method of step 2 new and P new The training function value set u new ; Based on the old true posterior distribution p(f|y,θ), use Bayes’ theorem to derive the new true posterior distribution p(f|y,y new ,θ new ); define the new approximate posterior distribution of the training function f as q new (f|θ new ), define the new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θ new ) and construct a new variational free energy bound based on the new KL divergence and the variational inference principle. According to u and u new The relationship between the two is used to derive the new variational free energy bound. About u, u new The expression of Select multiple different pseudo points P new , using numerical iteration method to calculate u new Perform iterative calculations to make the new approximate posterior distribution q new (f|θ new ) approximates the new true posterior distribution p(f|y,y new ,θ new )make Maximize and obtain the new optimal variational free energy bound and the new optimal approximate posterior distribution q new,vfe (u new |θ new );
[0140] It should be noted that in step 5, based on the old true posterior distribution p(f|y,θ) and the old optimal approximate posterior distribution q(f|θ), using Bayes’ theorem, the new true posterior distribution p(f|y,y new ,θ new ) According to formula (5.1), we can get:
[0141]
[0142] Where: q new (f|θ new ) is the new approximate posterior distribution after obtaining new monitoring data; p(f|y,y new ,θ new ) is the new true posterior distribution of the training function f after obtaining new monitoring data; is dependent on the hyperparameter θnew The standardization constant of ; To obtain the new monitoring data after the training function f new true prior distribution; p (y | f) to obtain the new monitoring data given the training function f in the new data under the conditional distribution of the old output vector y; p (y new |f) is the new output vector y in the new data given the training function f after obtaining the new monitoring data new The conditional distribution of
[0143] Among them, p(y|f) is obtained according to formula (5.2):
[0144]
[0145] Where: is a normalization constant that depends on the hyperparameter θ; q vfe (f|θ) is the optimal approximate posterior distribution before acquiring new monitoring data; To obtain the true prior distribution of the training function f before new monitoring data.
[0146] It should be noted that in step 5, the new variational free energy bound According to formula (5.3), we can get:
[0147]
[0148] Where: logp(y new |y,θ new ) is the new output vector y new The log-marginal likelihood function under the condition of the old output vector y; KL[q new (f|θ new )||p(f|y,y new ,θ new )] is the new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θ new ) new KL divergence between ;
[0149] Among them, KL[q new (f|θ new )||p(f|y,y new ,θ new )] According to formula (5.4), we can get:
[0150]
[0151] Where: p(f ≠u,new |u new ,θ new ) is the training function value set u except the new pseudo pointnew The remaining potential function value f ≠u,new Prior distribution of q new (u new |θ new ) is the new pseudo-point training function value set u new The posterior distribution of p(u new |θ new ) New pseudo-point training function value set u new Prior distribution of f ≠u,new =f new -u new ;f new is the set of training function values for the new historical monitoring data;
[0152] In this case, it is assumed that q(f|θ) obeys the normal distribution, that is, q(f|θ) is determined according to formula (5.5):
[0153]
[0154] Where: and are the mean and variance of the normal distribution respectively;
[0155] Among them, the new approximate posterior distribution q new (f|θ new ) According to formula (5.6), we can get:
[0156] q new (f|θ new )=q new (u new |θ new )p(f ≠u,new |u new ,θ new )(5.6)
[0157] Where: p(f ≠u,new |u new ,θ new ) is the training function value set u except the new pseudo point new The remaining potential function value f ≠u,new Prior distribution of q new (u new |θ new ) is the new pseudo-point training function value set u new The posterior distribution of .
[0158] It should be noted that in step 5, the new optimal approximate posterior distribution q new,vfe (f|θ new ) and the new optimal variational free energy bound According to formula (5.7) and (5.8) respectively, we can obtain:
[0159]
[0160] Where: p new (u new |θ new ) is the new pseudo point P new The training function value set u new The prior distribution of K fu,new f and u new The covariance matrix between uu,new for u and u new The covariance matrix between ; K uu,new The inverse matrix of
[0161] Step 6: According to Get θ new Value, according to q new,vfe (u new |θ new ) and θ new It is worth getting the new specific expression and variance of the training function f Substitute the new specific values into step 1 to obtain a new specific expression of the probability prediction model.
[0162] In the first step of the present invention, the standard Gaussian process model is used as the benchmark probability prediction model. In the second step, the monitoring data set is simplified by introducing pseudo points to construct an approximate probability prediction model based on the flow sparse Gaussian process. In the third step, the variational free energy bound is constructed by variational inference, and the optimal variational free energy bound is solved by numerical iteration. and the best approximate posterior distribution q vfe (f|θ), after solving the hyperparameter θ and the training function f, the two are used as the true solution of the baseline probability prediction model, so that the specific expression of the probability prediction model can be obtained. Based on the specific expression, the support displacement in the future period can be predicted, and the predicted support displacement is compared with the measured support displacement. According to the difference between the two, the support operation status can be evaluated; however, as time goes by, the model error will gradually increase, and the accuracy of the support operation status evaluation will decrease. Therefore, in step five of the present invention, the probability prediction model is updated based on the newly acquired monitoring data based on the new monitoring data, and a new specific expression of the probability prediction model based on the new historical monitoring data is established. The support displacement in the next time period is predicted based on the new expression. Compared with the old expression of the probability prediction model, the new expression of the probability prediction model contains the characteristics of the new monitoring data, and its prediction result is more accurate; and so on, new data is continuously acquired, and the prediction result of each stage is more reliable.
[0163] The present invention deploys a stream sparse Gaussian process approximation framework based on variational inference in a data stream environment where monitoring data arrives sequentially. According to the approximate posterior distribution of the training function f obtained based on the old historical monitoring data of the previous time period and the new approximate posterior distribution of the training function f obtained by updating the new historical monitoring data of the new time period, the storage of historical monitoring data in the stream sparse Gaussian process model, the updating of the stream sparse Gaussian process model and the real-time probability prediction of support displacement are realized, thus overcoming the problem of time-varying prediction model parameters caused by support aging and changes in the service environment.
[0164] The present invention only needs to call the approximate posterior distribution of the training function f in the previous time period, and can propagate the approximate posterior distribution forward without calling massive old historical monitoring data, which significantly reduces the memory requirement for storing massive data. At the same time, it can represent the inherent laws of historical monitoring data through a small number of pseudo points, while ensuring the model's prediction accuracy, further improving the model's online computing efficiency.
[0165] The applicant further states that while the present invention illustrates the implementation methods of the present invention through the above-described embodiments, the present invention is not limited to the above-described implementation methods. This does not mean that the present invention must rely on the above-described methods and structures for implementation. Those skilled in the art should understand that any improvements to the present invention, equivalent replacements for the implementation methods selected for the present invention, additions to steps, and selections of specific methods, etc., fall within the scope of protection and disclosure of the present invention.
[0166] The present invention is not limited to the above-mentioned embodiments. All embodiments that use structures and methods similar to those of the present invention to achieve the purpose of the present invention are within the protection scope of the present invention.
Claims
1. A bridge support displacement online probability prediction method, characterized by: The steps include: Step 1: Use the standard Gaussian process model as the baseline probability prediction model: y n =f(x n )+ε n , where ε n is the noise term, is the variance, f is the training function, and the training function f obeys the Gaussian process prior distribution with mean 0 and covariance k: Where θ is the hyperparameter on which the covariance k depends and The set of hyperparameters composed of Assume that the given training function f and variance After that, when the main beam temperature input vector is X, the support displacement output vector of the probability prediction model is y; Step 2: Obtain historical monitoring data of main beam temperature and historical monitoring data of bearing displacement according to Introducing a set of pseudo points in u i =f(z i ), M<N, M is the number of pseudo points, N is the number of samples; Definition: The set of training function values of pseudo point P The training function value set of historical monitoring data The remaining training function value set f ≠u =fu; According to u and f, f ≠u The relationship between the product rule is used to obtain the true prior distribution of the training function f About the expansion of u, and according to Get the expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u, and get the expansion of the approximate joint distribution p(f,y|θ) of the training function f and the output vector y with respect to u based on the original likelihood function of p(f|θ) and the output vector y; Define the true prior distribution of u under the output vector y as p(u|y,θ). According to p(u|y,θ) and the relationship between u and f, f ≠u The product rule is used to obtain the expansion of the true posterior distribution p(f|y,θ) of the training function f under the output vector y with respect to u, and the expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained based on p(f|y,θ); Step 3: Define the marginal likelihood function of the output vector y as p(y|θ), and construct the lower bound of the logarithmic marginal likelihood function logp(y|θ) based on the approximate posterior distribution q(f|θ) and the approximate joint distribution p(y,f|θ); define the KL divergence between the approximate posterior distribution q(f|θ) and the true posterior distribution p(f|y,θ), and construct the variational free energy bound based on the difference between the logarithmic marginal likelihood function logp(y|θ) and the KL divergence according to the variational inference principle. As a lower bound for logp(y|θ), we get Expressions about u; according to Select different pseudo points P and use numerical iteration method to iterate u, so that the approximate posterior distribution q(f|θ) approaches the true posterior distribution p(f|y,θ) so that Maximize and obtain the optimal variational free energy bound and the best approximate posterior distribution q vfe (f|θ); Step 4: According to Get the value of θ, according to q vfe (f|θ) and θ values to obtain the specific expression of f and Substitute the specific values of into step 1 to obtain the specific expression of the benchmark probability prediction model; Step 5: Get new monitoring data of main beam temperature and support displacement, and compare the new monitoring data with the old historical monitoring data in step 2. Composing new historical monitoring data according to Introduce a new pseudo point P according to the method of step 2 new and P new The training function value set u new ; Based on the old true posterior distribution p(f|y,θ), use Bayes’ theorem to derive the new true posterior distribution p(f|y,y new ,θ new ); Define the new approximate posterior distribution of the training function f as q new (f|θ new ), define the new approximate posterior distribution q new (f|θ new ) and the new true posterior distribution p(f|y,y new ,θ new ) and construct a new variational free energy bound based on the new KL divergence and the variational inference principle. According to u and u new The relationship between the two is used to derive the new variational free energy bound. About u, u new Expressions of according to Select multiple different pseudo points P new , using numerical iteration method to calculate u new Perform iterative calculations to make the new approximate posterior distribution q new (f|θ new ) approximates the new true posterior distribution p(f|y,y new ,θ new )make Maximize and obtain the new optimal variational free energy bound and the new optimal approximate posterior distribution q new,vfe (u new |θ new ); Step 6: According to Get θ new Value, according to q new,vfe (u new |θ new ) and θ new The new specific expression and variance of f are obtained Substitute the new specific values into step 1 to obtain a new specific expression of the baseline probability prediction model.
2. The method according to claim 1, wherein: In step 2, The true prior distribution of the training function f The expansion of u is obtained by using the product rule according to formula (2.1): Where: p(u|θ) is the prior distribution of u; p(f|u,θ) is the prior distribution of f under u; p(f ≠u |f,u,θ) is f ≠u Prior distribution under the conditions of f and u; Among them, p(u|θ) is determined according to formula (2.2): Where: K uu is the covariance matrix between u and u; Among them, p(f|u,θ) is determined according to formula (2.3): Where: K fu is the covariance matrix between f and u; K ff is the covariance matrix between f and f; K uf is the covariance matrix between u and f; K uu The inverse matrix of .
3. The method according to claim 2, wherein: In step 2, The expansion of the approximate prior distribution p(f|θ) of the training function f with respect to u is obtained according to formula (2.4): p(f|θ)=p(u|θ)q(f|u,θ)p(f ≠u |u,θ) (2.4) Where: q(f|u,θ) is the approximate posterior distribution of f under u, which is an approximation of p(f|u,θ); p(f ≠u |u,θ) is the prior distribution p(f ≠u |f,u,θ) is simplified.
4. The method according to claim 3, wherein: The approximate joint distribution p(y,f|θ) of the training function f and the output vector y is obtained according to formula (2.5): Where: is the original likelihood function; Substituting equation (2.4) into equation (2.5), the expansion of the approximate joint distribution p(y,f|θ) with respect to u is obtained according to equation (2.6): p(y,f|θ)=p(y|f,θ)p(f ≠u |u,θ)p(u|θ) (2.6) Where p(y|f,θ) is the prior distribution of y under the condition of f.
5. The method according to claim 4, wherein: In step 2, The true posterior distribution p(f|y,θ) of the training function f under the output vector y is expanded with respect to u according to formula (2.7): p(f|y,θ)=p(u|y,θ)p(f ≠u |y,u,θ) (2.7) Where p(u|y,θ) is the prior distribution of u under the condition of y; p(f ≠u |y,u,θ) is f ≠u Prior distribution under u and y conditions; The expansion of the approximate posterior distribution q(f|θ) of the training function f with respect to u is obtained according to formula (2.8): q(f|θ)=q(u|θ)p(f ≠u |u,θ) (2.8) Where q(u|θ) is the posterior distribution of u.
6. The method according to claim 5, wherein: In step 3, the lower bound of the log-marginal likelihood function logp(y|θ) is constructed according to formula (3.1): Where: logp(y|θ) is the logarithmic marginal likelihood function of y; is the variational free energy bound that depends on q(f|θ); Among them, the variational free energy bound Determine according to formula (3.2): Where: KL[q(f|θ)||p(f|y,θ)] is the KL divergence between q(f|θ) and p(f|y,θ); p(f|y,θ) is the true posterior distribution of f under y; Substituting equations (2.6) and (2.8) into equation (3.1), The expression of u is obtained according to formula (3.3): Where: f n =f(x n ) is x n The implicit function value at ; To eliminate items.
7. The method according to claim 6, wherein: In step 3, the optimal variational free energy bound F vfe (θ) and the best approximate posterior distribution q vfe (f|θ) is determined according to formula (3.4) and (3.5) respectively: Where: I is the unit matrix; trace(·) is the regularized tracking term.
8. The method according to claim 7, wherein: In step five, The new true posterior distribution p(f|y,y new ,θ new ) According to formula (5.1), we can get: Where: q new (f|θ new ) is the new approximate posterior distribution of f after acquiring new monitoring data; p(f|y,y new ,θ new ) is the new true posterior distribution of f after obtaining new monitoring data; is dependent on the hyperparameter θ new The normalization constant of p(f|θ new ) is the new true prior distribution of f after obtaining the new monitoring data; p(y|f) is the conditional distribution of the old output vector y in the new data given f after obtaining the new monitoring data; p(y new |f) is the new output vector y in the new data given f after obtaining new monitoring data new The conditional distribution of Among them, p(y|f) is obtained according to formula (5.2): Where: is a normalization constant that depends on the hyperparameter θ; q vfe (f|θ) is the optimal approximate posterior distribution of f before acquiring new monitoring data; To obtain the true prior distribution of f before new monitoring data.
9. The method according to claim 8, wherein: In step 5, the new variational free energy bound According to formula (5.3), we can get: Where: logp(y new |y,θ new ) is the new output vector y new The log-marginal likelihood function under the condition of the old output vector y; KL[q new (f|θ new )||p(f|y,y new ,θ new )] is q new (f|θ new ) and p(f|y,y new ,θ new ) new KL divergence between ; Among them, KL[q new (f|θ new )||p(f|y,y new ,θ new )] According to formula (5.4), we can get: Where: p(f ≠u,new |u new ,θ new ) is the remaining training function value f ≠u,new Prior distribution of q new (u new |θ new ) is the new pseudo point P new The training function value set u new The posterior distribution of p(u new |θ new ) is u new Prior distribution of f ≠u,new =f new -u new ;f new is the set of training function values for the new historical monitoring data; Where q(f|θ) is determined according to formula (5.5): Where: and are the mean and variance of the normal distribution respectively; Among them, the new approximate posterior distribution q new (f|θ new ) The expansion of u is obtained according to formula (5.6): q new (f|θ new )=q new (u new |θ new )p(f ≠u,new |u new ,i new (5.6) Where: q new (u new |θ new ) is u new The posterior distribution of .
10. The method according to claim 9, wherein: In step 5, the new optimal approximate posterior distribution q new,vfe (f|θ new ) and the new optimal variational free energy bound According to formula (5.7) and (5.8) respectively, we can obtain: Where: p new (u new |θ new ) is the new pseudo point P new The training function value set u new The prior distribution of K fu,new f and u new The covariance matrix between uu,new for u and u new The covariance matrix between ; K uu,new The inverse matrix of